The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: Pascal

When we talk about mathematical discoveries certain names are mentioned. These are names like Pascal, Euclid, Fermat, and Euler. These people become our mathematician Heroes. In our eyes, we often believe they pioneered the study. When we hear names like Mo Jing and Yang Hui in western society, most of us probably don’t even think anything of them.

But did you know that many of the great mathematical discoveries made in Western Mathematics were also made by Chinese mathematicians? In fact some mathematical discoveries we attribute to western mathematicians were even made by Chinese mathematicians far before they were discovered in the west.

I bring this up not necessarily to shame western culture, but because I find it fascinating. We have two cultures that really didn’t intermix ideas and traditions, yet it seems that they have made many similar mathematical discoveries. In my opinion these similarities in a way show that two totally different cultures with cast differences still have profound similarities that can unite them.

Also in the great debate of whether math is manmade or discovered I personally believe the similarity between western and Chinese mathematics is a point for Team Discovered. That might be only because I currently am on Team Discovered, though. I believe this is a point for Team Discovered because I feel if two separate cultures that are not trading ideas come up with the same mathematical truths then maybe they discovered them instead of just happened to share the same inventive thoughts. Still maybe this is the exact reason I should join Team Invention and I am just not thinking through my argument all the way.

Let’s talk about some of the similar discoveries in Chinese mathematics and western mathematics. Let’s try to focus on the person behind a concept that both cultures discover/invented. If feasible we should mention when the discovery/invention came about and how. Also how did it influence mathematics and the human race? I won’t focus on it, but you might even want to see if what we discuss puts you on team invention or team discovery for mathematics.

I guess the first Chinese work that I would like to point out is more of a compilation of Chinese works than the work of one individual, but did you know that of book very much like Euclid’s Elements existed in China? This book was the canon of a group of people called Mo Jing. They were the followers of Mozi and the canon contained, among philosophical insights, works on geometry. Mozi was actually a Chinese philosopher, but his teachings inspired his followers to consider mathematics as well. In fact this book contained a definition for a point similar to Euclid’s. To be specific, “a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it.” (http://history.cultural-china.com/en/167H8715H13206.html)

Now, one of the most famous mathematical discoveries is “Pascal’s Triangle.” “Pascal’s Triangle,” is a fascinating work. To describe it you build it from the top down. Put a one at the top. Build the triangle down adding 1 more number in each row. The value of the number below is the sum of the two numbers directly above it. If it is an edge case the number is 1.

Image: Drini and Conrad.Irwin, via Wikimedia Commons.

This discovery, made by Pascal while, through letters, he was exploring probability with Fermat, was also discovered much earlier by a Chinese man named Yang Hui. Even before Yang Hui it was described earlier by Jia Xian in 1100. Yang Hui in his book attributes the triangle to Jia and acknowledged that it was through this triangle that he found square roots as well as cubed roots.

I feel it is also very important that we discuss the book Zhou Bi Suan Jing. This book, which is a collection of anonymous works, contains one of the first proofs of the well-known and widely used Pythagorean Theorem. As a refresher, this a2+b2=c2. Controversy overshadows the actual date of the book which is assumed to be around 1046-256 BCE.

We can clearly see that mathematical ideas are not monopolized by western tradition. In fact, in my studies of Chinese mathematics, I found references Pascal’s Triangle being found in India and Iran. Pascal was a genius, but clearly he was not the original discoverer of the triangle that bears his name. Mathematics is a global study, applied in many ways similarly by many cultures.

Take some time and identify a culture. Make sure it is a culture that is so different from your own that, in a history class, this culture would study completely different things than what you studied. Now take what you know of your culture and the culture you chose and find similarities. Sometimes this can be hard. There are similarities such as in many cultures families eat together, but there are also many differences. What I am saying here is that in many ways math can be one of those similarities. This is neat! Math is as much a western study as it is an eastern study.

So next time you are learning about a western mathematician and how awesome he/she is, take some time and ask yourself if maybe the same ideas were explored by someone else in a different time in a different part of the world. Maybe even look it up. You might be surprised by what you find.

Today we are going to be looking into the different patterns in Pascal’s Triangle. I am not talking about Pascal from the Walt Disney Move Tangled. I am talking about Blaise Pascal, a famous French mathematician and philosopher. He was born on June 19, 1623, in Clermont-Ferrand, France. His mother died we he was just three years old, leaving behind his two sisters, his father, and himself. His father, Etienne Pascal, never remarried; instead he focused on educating his children, especially his son, Blaise Pascal. In 1653, Pascal released Traite du triangle arihmetique, which talked about binomial coefficients. This later became famous and became known as Pascal’s Triangle.

Image: Hersfold (public domain), via Wikimedia Commons.

Even though Blaise Pascal was the one to get all the credit for the triangle, the ancient Chinese actually developed it. The reason why he receives the credit for the triangle is because he discovered the patterns that are in the triangle. It could also be because the Europeans did not know about the previous discovery from China, and we have an Eurocentric math culture so we know it as Pascal’s Triangle. Before we go into the different patterns that he discovered, we are going to review how Pascal’s Triangle is made. It starts with the number one at the very top, this is called row 0. Row 1 consists of 1 and 1 and the next row consists of the numbers 1, 2, and 3. This is determined by adding the two numbers above to the left and to the right to get the coefficients in the row. For example, in row 2 to get the first number we add 0+1=1, 1+1=2, 1+0=1. You will do this process for each of the rows to get the different coefficients.

Now that we know more about what Pascal’s Triangle lets look at the many different patterns that are present in Pascal’s Triangle. The first one is called the Hockey Pattern. If you start at any 1 in the triangle and go diagonally until you choose to stop, the sum of all the numbers in that diagonal is the number just below the last number, ensuring that you are looking at the number below and to the opposite side that the diagonal would have continued. If you start on the left side of the triangle you will go down to the right diagonally, then when you want to stop, you will go below and to the left of the last number to find the sum of the numbers within that diagonal. For the right side, you will go below and to the right of the diagonal for the sum. For example, look at the highlight red to see the pattern. 1+6+21+56=84.

Two more patterns that are present in Pascal’s Triangle are called The Sum of Rows and Prime numbers. In the pattern, The Sum of Rows, the website All You Ever Wanted to Know About Pascal’s Triangle and More states: “The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row.” This is saying if you pick the row 5 the sum of all the numbers in that row would equal 25. Remember that the first row is considered row 0. The next pattern, Prime numbers, states that if the second number in a row is prime, then all the other numbers in that row are divisible by that number.

These are just a few of the patterns that are present in Pascal’s Triangle. There are many more patterns and I encourage you to do some more research and discover all the patterns in Pascal’s Triangle.